15 plies is achievable:
1. h4 d6 2. f3 Qd7 3. Nh3 Qxh3 4. Kf2 Qxh1 5. h5 Qh4+ 6. Kg1 Qg3 7. Kh1 Qh2+ 8. Kxh2
Similarly I also only found a 15 half move solution for
1. d3 Nc6 2. Bh6 Nb4 3. Kd2 Nxc2 4. Kc1 Nxa1 5. Nd2 Nc2 6. Kb1 Nb4 7. Ka1 Nxa2 8. Kxa2
Let's consider the latter and see if we can do better. If we could, white would only have 7 moves.
The obvious one first: It needs to be the white king. The black king would need 8 moves just to do
Ka1a2 on an otherwise empty board.
If we ignore castling for now, the white king has to move 4 times to reach a1 and once more to capture on a2.That only leaves 2 of white's moves to spare. On its way to a1 at least four pieces - the a1 rook and the piece or pawn on the b, c, and d file - either have to move out of the way or have to be captured. Since we established white has only 2 non-king moves this means black has to capture at least two of them and additionally land on a2, all within 6 moves. Can black actually use 2 pieces for that? I don't think so. If black used 2 different pieces, one has to survive and make it to a2. But the fastest black can land any piece on a2 is in 3 moves, namely with the h2 knight. In the remaining 3 moves black can only do one capture tho, since the distance is too large for knights and for queen/bishop one move has to be spent to open up the piece, one move to line up the diagonal (since neither bishop or the queen target anything on the a-d files), which leaves only one capture which we have determined is not enough to help white.
So we determined that black can only infiltrate with one piece.
Further, notice that if white were to clear a1 by themself they need to be careful not to block the path of their King again, i.e., not leave the rook on the back rank which they've spent a move to clear. But also moving the rook along the a-file takes two moves since the a-pawn would have to move first. There is the possibility that black clears the a-pawn, then moves out of the way, white plays Ra2 and black captures on e2 but this simply takes too much time since black needs 3 moves at least to capture on a2 in the first place.
This means the one black piece doing the infiltration must capture the rook and end on a2 at the same time. The only piece that can do this is the queen.
Notice that the first capture (that clears the way for the king) can only occur on black's third move. Also white may not capture the queen afterwards since we already determined one piece has to do all the infiltration and land on a2 at the end. This essentially still blocks the square the queen captured on and even worse, it also blocks all adjacent squares. Lastly, black's queen must be on a2 after black's 6th move but somehow has to allow the white king to reach a1 on white's 6th move. It's quite clear that this implies that black's 5th move can not be in white's queenside since otherwise the queen would block the king's access to a1.
So to summarize (without casling):
- White has to move the king on all but 2 moves
- Black must land the queen on a2 on their 6th move
- Black must retreat the queen on their 5th move to allow the king to reach a1
- Black can not capture on their first 3 moves
The first and last statement imply that within white's first 3 moves they have to move their king and since black cannot capture they have to make way themself. This leaves only a d-pawn move into Kd2. As soon as the king is on d2 the a3 bishop may no longer be captured since the king would be forced to take black's queen (or run from his target square). This means white has to move the bishop out, essentially locking down the first 3 moves: d3 (or d4) - bh6 (or g5/f4/e3) - kd2. From here white would have to move the king on every move and remember that at this point black still would not have had the chance to capture. It's not too hard to see that this move sequence will not work.
So this leaves castling as our only option. But that does not work fast enough either. After castling, the king needs 3 to make more moves (including Ka1xa2) which leaves 3 moves to clear the back rank. However, clearing the back rank certainly requires 4 moves: d4 - qd3 - bd2 - nc3. Black can also not be of any help since even if they captured on the back rank (which they couldn't even do fast enough) and moved out, they'd cover the square on which they took, disallowing white to castle.
So long story short: I'm pretty sure
Ka1xa2 is only possible as the 15th half move and not earlier.